The extended metamorphosis of a complete bipartite design into a cycle system

Elizabeth J Billington
2004 Discrete Mathematics  
A Kt;t-design of order n is an edge-disjoint decomposition of Kn into copies of Kt;t. When t is odd, an extended metamorphosis of a Kt;t-design of order n into a 2t-cycle system of order n is obtained by taking (t − 1)=2 edge-disjoint cycles of length 2t from each Kt;t block, and rearranging all the remaining 1-factors in each Kt;t block into further 2t-cycles. The 'extended' refers to the fact that as many subgraphs isomorphic to a 2t-cycle as possible are removed from each Kt;t block, rather
more » ... han merely one subgraph. In this paper an extended metamorphosis of a Kt;t-design of order congruent to 1 (mod 4t 2 ) into a 2t-cycle system of the same order is given for all odd t ¿ 3. A metamorphosis of a 2-fold Kt;t-design of any order congruent to 1 (mod t 2 ) into a 2t-cycle system of the same order is also given, for all odd t ¿ 3. (The case t = 3 appeared in Ars Combin. 64 (2002) 65-80.) When t is even, the graph Kt;t is easily seen to contain t=2 edge-disjoint cycles of length 2t, and so the metamorphosis in that case is straightforward.
doi:10.1016/j.disc.2003.11.025 fatcat:5ie4pvz6wvcqdflujx6j36jjou